Georgia Institute of Technology College of Sciences

In this talk, I discuss our recent proof that there is a uniform bound forthe number of rational points on genus g curves of Mordell-Weill rank atmost g-3, extending a result of Stoll on hyperelliptic curves. I outlinethe Chabauty-Coleman for bounding the number of rational points on a curveof low Mordell-Weil rank and discuss the challenges to making the bounduniform. These challenges involving p-adic integration and Newton polygonestimates, and are answered by employing techniques in Berkovich spaces,tropical geometry, and the Baker-Norine theory of linear systems on graphs.